As you have described the carbon arc light, it seems to be a self regulating, constant current device, with the arc in series with the coil. As voltage increases, the arc length should increase as well, providing differing levels of brightness.
Old carbon arc movie projectors had some small servo motors and carbon rods on an eccentric to move the electrodes nearer to each other as they burned away. This was to maintain constant brightness. The old carbon arc streetlights also had some sort of mechanism to regulate brightness.
But I see none of that in looking quickly at the inside of the headlamp. OFF - the electrodes are closed, and ON they separate to a predetermined stop. If there is a coil in the lamp creating a magnetic field, and you mention these as constant current devices, then that field would be constant and not able to regulate arc length.
The nameplate on the light shows patent dates around 1902 and is stamped with 4 AMPS power consumed. No rated voltage, other than the full line voltage. Would that have operated in BRIGHT mode at the full 550 (600) volts?
This is very interesting as I'd not heard of carbon arcs running at such low current and high voltage.
I can tell you a bit about movie theatre and spotlight lamps. I don't know how much is applicable here but maybe it will be a bit helpful.
The movie theatre and spotlight arc lamps were always high current, low voltage DC, produced by an MG set (or later on by a transformer / silicon diode rectifier). There might be about 20-something volts across the arc so even a small 1kW lamp would be running at 50A.
The power source was always such that the voltage dropped with greater current. In cases where the utility actually supplied DC they'd have a big resistor bank in series with each lamp. That provided the drooping voltage characteristic automatically.
Those lamps required manually touching the rods to strike the arc with the feed automatic after that (though on some lamps a lot of manual fussing might still be needed). On a projector lamp there is a large reflector and the positive faces backward away from the projector and towards the reflector.
On a spotlight it's typically opposite with the positive, the arc crater of which is where the bulk of the light comes from, facing forward to a huge condenser lens.
In both cases, the position of the positive relative to the condenser lens (on a spotlight) or reflector (projection lamp) is what is critical. The position of the negative only matters as to gap width.
Lamps had drive motors to drive the rods towards each other in proportion to them being consumed. The motor was wired in a way that if the arc gap increased, the voltage across it increased and the motor, being wired across the gap, sped up and drove the rods closer which brought the voltage back down, providing some degree of self-regulation. A rheostat let you adjust the speed to try to keep the positive in position and the gap proper.
Most lamps had a tiny pinhole, mirror and screen on the side projecting an image of the rods so one could try to keep them aligned to marked ideal positions.
A simple lamp would have a single motor and drove rods via a screw with opposing threads and the positive part of the screw had a greater pitch since it was consumed faster but the ratio was fixed. Other lamps drove the positive via a screw and moved the negative in little increments via a mechanism so that motor speed affected both but the amount of advance of the negative relative to positive could be adjusted. Other, fancier lamps had separate motors and rheostats for each rod.
The whole idea was to be able to get things into good adjustment so little manual attention was needed over the course of a 20 minute film reel. (Theatres had two projectors switching back and forth as the movie progressed. Projector changeovers went unnoticed by most people.)
Some late model arc lamps could run a full hour uninterrupted--enough for a large 24" film reel holding 3 regular reels worth of film.
I hope some of this may be helpful. BTW, the xenon lamps now in use are also high current, low voltage DC arc devices. Since the electrodes in the bulbs cannot move, they are struck via a high voltage RF zap from an ignitor circuit in the lamphouse.